3.48.1 \(\int \frac {240 x+768 x^4}{25-320 x^3+64 e^2 x^4+1024 x^6+e (80 x^2-512 x^5)+(-80 x^2-128 e x^4+512 x^5) \log (4)+64 x^4 \log ^2(4)} \, dx\)

Optimal. Leaf size=21 \[ -\frac {3}{-e+4 \left (-\frac {5}{32 x^2}+x\right )+\log (4)} \]

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Rubi [A]  time = 0.24, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 5, integrand size = 74, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6, 1593, 6688, 12, 1588} \begin {gather*} \frac {24 x^2}{-32 x^3+8 x^2 (e-\log (4))+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(240*x + 768*x^4)/(25 - 320*x^3 + 64*E^2*x^4 + 1024*x^6 + E*(80*x^2 - 512*x^5) + (-80*x^2 - 128*E*x^4 + 51
2*x^5)*Log[4] + 64*x^4*Log[4]^2),x]

[Out]

(24*x^2)/(5 - 32*x^3 + 8*x^2*(E - Log[4]))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {240 x+768 x^4}{25-320 x^3+1024 x^6+e \left (80 x^2-512 x^5\right )+\left (-80 x^2-128 e x^4+512 x^5\right ) \log (4)+64 x^4 \left (e^2+\log ^2(4)\right )} \, dx\\ &=\int \frac {x \left (240+768 x^3\right )}{25-320 x^3+1024 x^6+e \left (80 x^2-512 x^5\right )+\left (-80 x^2-128 e x^4+512 x^5\right ) \log (4)+64 x^4 \left (e^2+\log ^2(4)\right )} \, dx\\ &=\int \frac {48 x \left (5+16 x^3\right )}{\left (5-32 x^3+8 x^2 (e-\log (4))\right )^2} \, dx\\ &=48 \int \frac {x \left (5+16 x^3\right )}{\left (5-32 x^3+8 x^2 (e-\log (4))\right )^2} \, dx\\ &=\frac {24 x^2}{5-32 x^3+8 x^2 (e-\log (4))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.29 \begin {gather*} -\frac {24 x^2}{-5-8 e x^2+32 x^3+8 x^2 \log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(240*x + 768*x^4)/(25 - 320*x^3 + 64*E^2*x^4 + 1024*x^6 + E*(80*x^2 - 512*x^5) + (-80*x^2 - 128*E*x^
4 + 512*x^5)*Log[4] + 64*x^4*Log[4]^2),x]

[Out]

(-24*x^2)/(-5 - 8*E*x^2 + 32*x^3 + 8*x^2*Log[4])

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fricas [A]  time = 0.59, size = 28, normalized size = 1.33 \begin {gather*} -\frac {24 \, x^{2}}{32 \, x^{3} - 8 \, x^{2} e + 16 \, x^{2} \log \relax (2) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((768*x^4+240*x)/(256*x^4*log(2)^2+2*(-128*x^4*exp(1)+512*x^5-80*x^2)*log(2)+64*x^4*exp(1)^2+(-512*x^
5+80*x^2)*exp(1)+1024*x^6-320*x^3+25),x, algorithm="fricas")

[Out]

-24*x^2/(32*x^3 - 8*x^2*e + 16*x^2*log(2) - 5)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((768*x^4+240*x)/(256*x^4*log(2)^2+2*(-128*x^4*exp(1)+512*x^5-80*x^2)*log(2)+64*x^4*exp(1)^2+(-512*x^
5+80*x^2)*exp(1)+1024*x^6-320*x^3+25),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.27, size = 28, normalized size = 1.33

method	result	size
risch	\(\frac {3 x^{2}}{x^{2} {\mathrm e}-2 x^{2} \ln \relax (2)-4 x^{3}+\frac {5}{8}}\)	\(28\)
gosper	\(\frac {24 x^{2}}{8 x^{2} {\mathrm e}-16 x^{2} \ln \relax (2)-32 x^{3}+5}\)	\(29\)
norman	\(\frac {24 x^{2}}{8 x^{2} {\mathrm e}-16 x^{2} \ln \relax (2)-32 x^{3}+5}\)	\(29\)
default	\(\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (25+1024 \textit {\_Z}^{6}+\left (-512 \,{\mathrm e}+1024 \ln \relax (2)\right ) \textit {\_Z}^{5}+\left (-256 \,{\mathrm e} \ln \relax (2)+256 \ln \relax (2)^{2}+64 \,{\mathrm e}^{2}\right ) \textit {\_Z}^{4}-320 \textit {\_Z}^{3}+\left (80 \,{\mathrm e}-160 \ln \relax (2)\right ) \textit {\_Z}^{2}\right )}{\sum }\frac {\left (16 \textit {\_R}^{4}+5 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3} {\mathrm e}^{2}-32 \,{\mathrm e} \ln \relax (2) \textit {\_R}^{3}-80 \textit {\_R}^{4} {\mathrm e}+32 \textit {\_R}^{3} \ln \relax (2)^{2}+160 \textit {\_R}^{4} \ln \relax (2)+192 \textit {\_R}^{5}+5 \textit {\_R} \,{\mathrm e}-10 \ln \relax (2) \textit {\_R} -30 \textit {\_R}^{2}}\right )}{2}\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((768*x^4+240*x)/(256*x^4*ln(2)^2+2*(-128*x^4*exp(1)+512*x^5-80*x^2)*ln(2)+64*x^4*exp(1)^2+(-512*x^5+80*x^2
)*exp(1)+1024*x^6-320*x^3+25),x,method=_RETURNVERBOSE)

[Out]

3*x^2/(x^2*exp(1)-2*x^2*ln(2)-4*x^3+5/8)

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maxima [A]  time = 0.36, size = 26, normalized size = 1.24 \begin {gather*} -\frac {24 \, x^{2}}{32 \, x^{3} - 8 \, x^{2} {\left (e - 2 \, \log \relax (2)\right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((768*x^4+240*x)/(256*x^4*log(2)^2+2*(-128*x^4*exp(1)+512*x^5-80*x^2)*log(2)+64*x^4*exp(1)^2+(-512*x^
5+80*x^2)*exp(1)+1024*x^6-320*x^3+25),x, algorithm="maxima")

[Out]

-24*x^2/(32*x^3 - 8*x^2*(e - 2*log(2)) - 5)

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mupad [B]  time = 3.39, size = 27, normalized size = 1.29 \begin {gather*} \frac {24\,x^2}{-32\,x^3+\left (8\,\mathrm {e}-16\,\ln \relax (2)\right )\,x^2+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((240*x + 768*x^4)/(256*x^4*log(2)^2 - 2*log(2)*(128*x^4*exp(1) + 80*x^2 - 512*x^5) + exp(1)*(80*x^2 - 512*
x^5) + 64*x^4*exp(2) - 320*x^3 + 1024*x^6 + 25),x)

[Out]

(24*x^2)/(x^2*(8*exp(1) - 16*log(2)) - 32*x^3 + 5)

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sympy [A]  time = 1.50, size = 26, normalized size = 1.24 \begin {gather*} - \frac {24 x^{2}}{32 x^{3} + x^{2} \left (- 8 e + 16 \log {\relax (2 )}\right ) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((768*x**4+240*x)/(256*x**4*ln(2)**2+2*(-128*x**4*exp(1)+512*x**5-80*x**2)*ln(2)+64*x**4*exp(1)**2+(-
512*x**5+80*x**2)*exp(1)+1024*x**6-320*x**3+25),x)

[Out]

-24*x**2/(32*x**3 + x**2*(-8*E + 16*log(2)) - 5)

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